Question

The A.M. of 9 terms is 15 . If one more term is added to this series then the A.M. becomes 16 . The value of added term is (a) 30 (b) 27 (c) 25 (d) 23

Short answer

The value of the added term is 25. The correct answer is (c) 25.

Step-by-step solution

Find the sum of the original series

We know that the arithmetic mean of 9 terms is 15, so we can use the formula for the average: A.M. = (Sum of all the terms) / (Total number of terms) In this case, the average (A.M.) is 15, and the total number of terms is 9, so: 15 = Sum of all terms / 9 Now, let's find the sum of all terms: Sum of all terms = 15 * 9 = 135

Find the sum of the updated series with the added term

Now, we are given that the new A.M. becomes 16 when one more term is added to the series. As a result, there are now 10 terms in total. Using the A.M. formula once again: 16 = Sum of the updated series / 10 Now, let's find the sum of the updated series: Sum of the updated series = 16 * 10 = 160

Determine the value of the added term

Finally, we will find the value of the added term by subtracting the sum of the original 9 terms from the sum of the updated series of 10 terms: Added term = Sum of the updated series - Sum of the original series Added term = 160 - 135 Added term = 25 Therefore, the value of the added term is 25. The correct answer is (c) 25.

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference, denoted as 'd'. For example, in the sequence 2, 4, 6, 8, ..., each term increases by 2, so this is an arithmetic progression with a common difference of 2.

An arithmetic progression can be written generally as: \[a, a + d, a + 2d, a + 3d, ...\] where 'a' is the first term, and 'd' is the common difference. By understanding the structure of an arithmetic progression, one can easily compute various properties of the sequence, such as the sum of the first 'n' terms, the 'n'th term, and of course, the arithmetic mean.

A.M. Formula

The Arithmetic Mean (A.M.) is a measure of central tendency that indicates the average value of a set of numbers. The formula to calculate the arithmetic mean is quite straightforward:

\[ A.M. = \frac{{Sum \ of \ all \ terms}}{{Total \ number \ of \ terms}} \]
This formula is essential when dealing with sequences, particularly arithmetic progressions. In the context of the given problem, understanding how to use the A.M. formula is crucial for determining the overall average after the introduction of a new term to the sequence. This addition can alter the sum and consequently the mean. Thus, after finding the initial sum, the A.M. formula becomes instrumental in resolving what the final, added term must be to adjust the mean accordingly.

Sequence and Series

In mathematics, a sequence is a list of numbers in a specific order, whereas a series is the sum of the terms of a sequence. Sequences can be finite or infinite, and arithmetic sequences are just one type of sequence.

Each type of sequence or series has its own set of formulas and properties. For arithmetic series, which is the sum of terms in an arithmetic progression, there are specific formulas to calculate sums over a certain number of terms. Understanding these concepts helps with solving problems related to sums and averages of sequences and series. Illustrating these in real-world problems, like finding a new average when additional values are included, demonstrates the practical applications of sequences and series in analyzing and interpreting data.